[论文] [论文] Moore, Escher, Penrose: A Conformal Golden Braid

论文概要

研究领域: CV 作者: Sophia Feldman, Assaf Shocher 发布时间: 2026-10-01 arXiv: 2610.02210

中文摘要

M.C. 埃舍尔曾这样描述他1956年的石版画《画廊》："我一生中从未做过如此奇特的事情……一个年轻人正饶有兴趣地看着墙上的一幅展品，而展品中正是他自己。这怎么可能？"近半个世纪后，有数学分析将其几何结构与一幅未扭曲的源图像通过共形幂映射 z -> z^alpha 联系起来。在此构造基础上，我们使用冻结的文本到图像扩散模型生成新的自指涉场景。仅靠提示无法强制递归，后处理变换可能导致结构连接不良，在采样过程中施加变换也不够——去噪器可能"修复"预期的畸变或偏离规定几何。我们为非可逆图像变换 T 构造了适配其递归约束的广义逆 T^dagger。在理想化表述中，Penrose 恒等式 T*T^dagger*T = T 使 T*T^dagger 成为到几何可行图像集上的幂等投影。然而即使有了投影，仅对变换后图像去噪仍是分布外任务。因此我们将去噪步骤与 T 和 T^dagger 编织在一起：源空间步骤构建未扭曲场景，变换空间步骤精修其在最终几何中的外观和连接。我们生成了类似《画廊》的构图并探索了更多变换——不是扭曲一张已完成图像，而是让场景与其畸变共同生长。

原文摘要

I don't think I have ever done anything as peculiar in my life. Among other things, it shows a young man looking with interest at a print on the wall of an exhibition that features himself. How can this be? Perhaps I am not far removed from Einstein's curved universe.'' So wrote M.C. Escher about his 1956 lithograph Print Gallery. Nearly half a century later, a mathematical analysis related its geometry to an untwisted source image through a conformal power map \(z \mapsto z^α\), \(α\in \mathbb{C}\). Building on this construction, we use a frozen text-to-image diffusion model to generate new self-referential scenes. Prompting alone does not enforce the recursion, while a post-hoc transformation can leave structures poorly connected. Applying the transformation during sampling is also insuffici...

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